Joel David Hamkins: Philosophy of Mathematics and Truth
Paradigm
Incompleteness: Expanding Mathematical Understanding
This chapter explores the concept of incompleteness in mathematics, discussing its historical context, emotional perception, and the importance of incorporating new knowledge. It explains the second incompleteness theorem and the consistency hierarchy that arises from it, as well as the concept of creating a hierarchy in mathematics to prove consistency at lower levels. The chapter also addresses the role of axioms in mathematics, the distinction between intrinsic and extrinsic justification, and the debate surrounding the adoption of axioms.
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